1986). Unfortunately there is no simple linear combination of parameter that guarantees
concavity. This has usually been "tested" checking whether the substitution elasticities
matrix is semidefinite negative for each actual share. However concavity can be
imposed via Cholesky decomposition, as suggested by Lau (1978)2. Since the matrix of
the share elasticities Ω = [γ iJ is symmetric, than it is possible to represent it in terms of
its Cholesky factorization Ω = T’DT, where T is a unit lower triangular matrix and D is
a diagonal matrix. Symmetry and product exhaustion provide conditions under which
there exists a one to one transformation between the elements of ### Ω and the
elements of T and D. Then the matrix of share elasticities is semidefinite negative if and
only if the first N-1 diagonal elements of D are nonpositive. This procedure has been
first applied by Jorgenson and Fraumeni (1981), but it has been shown by Diewert and
Wales (1987) that it destroys the second order flexibility properties, since it implicitly
assumes "restrictions on own and cross elasticities that may be apriori completely
unacceptable ... the use of the Jorgenson-Fraumeni procedure for imposing concavity
will lead to estimated input substitution matrices which are in some sense "too negative
semidefinite"; i.e. the degree of input substitutability will tend to be biased in an upward
direction" (Diewert and Wales 1987, p. 48). Below we follow the bayesian approach to
impose locally concavity and monotonicity that doesn’t suffer this drawback.
Parameters and their transformations provide information about the technology.
In this paper we focus on price and substitution elasticity. Price elasticities can be easily
obtained by Allen's definition of elasticities of substitution:
C (∂ 2C / ∂pi∂pi)
σ, =------------—
ii
εγ
ii =ii +1 ,
yi yiyi
for i≠j
(2.4)
εij
σii ==1
γij
+ij
2,
(2.5)
while price elasticities are:
ε =yσ
(2.6)
ij yj ij
Since we want to keep the analysis as simple as possible I choose a value added
translog cost function that generates the following labor share equation:
2 A complete discussion about concavity can be found in Morey (1986).